*all page references are from this document unless

1

*All page references are from this document unless otherwise noted.

Design and Organization of the Common Core State Standards

(CCSS) for Mathematics •  Introduction

•  Standards for Mathematical Practice

•  Standards for Mathematical Content

•  Glossary

2

Introduction: Where was American education before CCSS?

(CCSS pp 3-4)

Poor Performance

Poorly Aligned Curriculum

Too many standards Weak Textbooks

3

Structure Sample from CCSS – Mathematics

( Refer to CCSS page 5)

4

Structure Sample from Grade 6 (Refer to CCSS page 45)

Statistics and Probability 6.SP

5

Develop understanding of statistical variability.

1.  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

2.  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

3.  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Standard Cluster

Domain

Cluster Heading (According to

PARCC)

Reading the Grade Level Standards

Standards define what students should understand and be able to do.

Clusters are groups of related standards.

*Note: standards from different clusters may sometimes be closely related.

Domains are larger groups of related standards.

*Note: standards from different domains may sometimes be closely related.

6

Domains for Grades 6-8

•  RP = Ratios and Proportional Relationships (6-7)

•  NS = The Number System (6-8)

•  EE = Expressions and Equations (6-8)

•  G = Geometry (6-8)

•  SP = Statistics and Probability (6-8)

•  F = Functions (8 only)

7

Referencing the CCSS for Mathematics

7.SP.1 7.SP.1 - Grade level 7.SP.1 - Statistics and Probability Domain 7.SP.1 - Standard

8.EE.7a 8.EE.7a - Grade level 8.EE.7a - Expressions and Equations Domain 8.EE.7a - Standard

8


(Refer to CCSS page 42)

What is the reference for the following standard?

“Use ratio reasoning to convert measurement units; manipulate and transform units appropriately

when multiplying or dividing quantities.”

Answer: 6.RP.3d

9


(Refer to CCSS page 42)

6.RP.3d •  Specify the grade level.

•  Identify the domain.

•  Identify the cluster heading.

•  Specify the standard number. 10

Grade 6

(RP) Ratios and Proportional Relationships

Understand ratio concepts and use ratio reasoning to solve problems.

3d

Algebra

Measurement

Ratios and Proportional Relationships

Expressions and

Equations

Functions

Numbers

Geometry

Statistics and Data

Mississippi Mathematics Framework (MMF) Content Strands 6-8 vs.

CCSS Domains 6-8

11 MMF 6-8 CCSS 6-8

Reviewing the CCSS for Mathematics Glossary

•  Locate pages 85-90 of the CCSS for Mathematics.

•  Note the following:  List of Terms (pp 85 – 87)  Table 1 (p. 88)  Table 2: (p. 89)  Tables 3, 4 and 5: (p. 90)

12

A Snapshot of the Glossary (Refer to List of Terms CCSS page 85)

Glossary Additive inverses. Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and – 3/4 are additive inverses of one another because 3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.

Associative property of addition. See Table 3 in this Glossary.

Associative property of multiplication. See Table 3 in this Glossary.

Bivariate data. Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team.

Box plot. A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.1

13

14

A Snapshot of the Glossary (Refer to Table 1 CCSS page 88)

15


Table 3. The properties of operations. Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.

16


Associative property of addition

Commutative property of addition

Additive identity property of 0

Existence of additive inverses

Associative property of multiplication

Commutative property of multiplication

Multiplicative identity property of 1

Existence of multiplicative inverses

Distributive property of multiplication over addition

(a + b) + c = a + (b + c)

a + b = b + a

a + 0 = 0 + a = a

For every a there exists –a so that a + (–a) = (–a) + a = 0.

(a × b) × c = a × (b × c)

a × b = b × a

a × 1 = 1 × a = a

For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1.

a × (b + c) = a × b + a × c

Work Session 1 CCSS for Mathematics

“Scavenger Hunt” Directions: • Locate Work Session 1 Activity Sheet.

• Knowing where to find information in the Standards is just as important as knowing the information itself. Using the CCSS for Mathematics, work in pairs to find the answers to the questions.

17


“Scavenger Hunt”

Directions: Facilitator will discuss answers for Work Session 1.

18


Directions: Using the Promethean Clickers on your table, respond to the following statement:

“When teaching the CCSS for Mathematics you must teach the

standards in the order they appear in the CCSS document.”

19

The Heart of the CCSS for Mathematics:

Standards for Mathematical

Practice 20

Standards for Mathematical Practice

(Refer to CCSS pp 6-8)

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

21


(Refer to CCSS pp 6-8)

1.  Make sense of problems and persevere in solving them.

2.  Reason abstractly and quantitatively.

3.  Construct viable arguments and critique the reasoning of others.

4.  Model with mathematics.

5.  Use appropriate tools strategically.

6.  Attend to precision.

7.  Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

22


The Standards for Mathematical Practice should not be used as a checklist nor should they be used in isolation. Rather, the Mathematical Practices should be interwoven into every lesson where they overlap and interact with each other constantly.

23


Provide connected, engaging

instruction

Practice with the content and

mature mathematically

Mathematical Practices

24

The Role of the Teacher

The Role of the Student

The Learning Environment

Work Session 2: Connecting Mathematical

Practices to Instruction Directions: •  Preview the Standards for Mathematical Practice

handout.

•  Locate the large “card” on your table. The number on the front indicates the Mathematical Practice your group will discuss.

•  Locate Work Session 2 Activity Sheet and complete this Activity as a group.

25

Work Session 2: Connecting Mathematical

Practices to Instruction

Directions: Facilitator will select several groups to discuss their responses for Work Session 2.

26

Unpacking the CCSS for Mathematics and

Creating Essential Questions

27

Steps for Unpacking CCSS for Mathematics

1.  “Study” the standard as a Professional Learning Community (PLC).

2.  Identify prerequisite skills.

3.  Identify the key terms and verbs directly (or indirectly) stated within the standard.

4.  Give a definition for each term and verb.

5.  Provide “student-friendly language” for each term and verb.

6.  Create a series of “I can” statements in “student friendly language”.

7.  Create a series of Essential Questions.

28

What is an Essential Question?

An Essential Question is a Question that: •  Causes genuine and relevant inquiry into the big ideas and core

content.

•  Provokes deep thought, lively discussion, sustained inquiry, and new understanding.

•  Requires students to consider alternatives, weigh evidence, support their ideas, and justify their answers.

•  Sparks meaningful connections with prior learning and personal experiences.

•  Naturally recurs, creating opportunities for transfer to other situations and subjects. 29

Why Do We Need Essential Questions?

•  Guides instruction for the teacher and the students.

•  Assists students in seeing the relevancy of a topic of study.

•  Serves as a framework to provide and sustain student interest.

•  Fosters a literacy and vocabulary-rich environment.

•  Links to other essential questions and topics.

•  Ensures utility of the Standards for Mathematical Practice.

30

Constructing Essential Questions

31

1.  “Study” the standard in your PLC. --Examine your teaching objectives and goals within the

standard.

--Identify the key words.

--Tie to prerequisite skills.

--Ask yourself “why” is this question important?

2.  Possibly write the standard as a question or a series of smaller questions.

3.  Think: concept → skill→ application → understanding

Unpacking Sample for CCSS 8.F.4

Directions: • Locate the Work Session 3 “Unpacking Sample for CCSS 8.F.4”.

• Facilitator will discuss the Unpacking Sample.

32

Work Session 3 Activity 3a: Unpacking CCSS for Grades 6, 7, or 8 and Creating Essential

Questions Directions: • Locate Work Session Activity Sheet 3a.

• In groups of 3, complete Activity Sheet 3a for one of the standards listed below.

CCSS 6.G.1 CCSS 7.G.6 CCSS 8.G.8

33

Work Session 3 Activity 3a continued

Directions: • Locate the large “card” on your table. The back of the card indicates which section from Activity 3a your group will record on chart paper.

• Upon completion, designate one person to post your work in the designated area and report out.

34


35


“The Unpacking Activity could be used by students, as well as teachers,

prior to teaching a unit.”

Work Session 3 Activity 3b: Instructional Strategy for CCSS 7.G.6

Directions: •  Locate Work Session 3 Activity Sheet 3b.

•  Complete model problem with the facilitator.

36

Work Session 3 Activity 3b continued

Directions: •  In groups of 3, select one geometric solid

from the formula chart or from the geometric solids on your table.

•  Complete the last row of the Activity Sheet.

37


Directions: •  Designate one person to show your work

on chart paper and post it.

•  Facilitator will select several groups to report out.

38

Work Session 4 Activity 4a: Focusing on a Grade 6 CCSS

(6.RP.2) Directions: •  Locate Work Session 4a Activity Sheet.

•  As a group, complete Work Session 4a Activity Sheet.

•  View the video.

39


(6.RP.2)

Directions: Facilitator will discuss answers for Work Session 4a.

40

Work Session 4 Activity 4b: Focusing on a Grade 6 CCSS

(6.RP.2) Directions: •  Facilitator will model alternative method for teaching

CCSS 6.RP.2.

•  Locate Work Session 4b Activity Sheet.

•  In groups of 3, choose Option A or B.

•  Complete problem #1 using the method modeled by the facilitator.

•  Complete problem #2 using the method modeled in the video. 41


(6.RP.2)

Directions: Facilitator will discuss answers for Work

Session 4b.

42

Example of Scaffolding and Progression in the CCSS

Grades 6 - 8 using 6.RP.2 8th Grade (8.EE.5) Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

7th Grade (7.RP.1) Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

6th Grade (6.RP. 2) Understand the concept of a unit rate a/b associated with a ratio a:b with b≠0 and use rate language in the context of a ratio relationship. 43


(7.EE.4a)

Directions: •  Locate Work Session 5a Activity Sheet.

•  Complete items #1-5 individually.


44

Work Session 5 Activity 5a: continued

(7.EE.4a)


Session 5a (items #1-5).

45


(7.EE.4a)


•  Complete items #7-8 as a group.

46


(7.EE.4a)

Directions: •  Facilitator will select several groups to

report out.

47


(7.EE.4a)

Directions: •  Locate Work Session 5b Activity Sheet.

•  Complete Work Session 5b Activity Sheet as a group using the method modeled in the video.

48

Directions: • Designate one person to show your work

for item #4 on chart paper and post it.

•  The facilitator will select several groups to discuss their response to item #4.

49


(7.EE.4a)


(7.EE.4a)


Session 5b (items #1-3).

50


(8.F.4)


•  Complete items #1-4.


51


(8.F.4)


Session 6a (items#1-4).

52


(8.F.4)


•  Complete item #6 as a group.

•  The facilitator will select several groups to discuss their response to item #6.

53


(8.F.4)


Session 6a (item #6).

54


(8.F.4)

Directions: •  Locate Work Session 6b Activity Sheet.

•  Complete Model Problem with facilitator.

55

Work Session 6 Activity 6c: Focusing on a Grade 8 CCSS

(8.F.4)

Directions: •  Locate Work Session 6c Activity Sheet.

•  As a group, complete Work Session 6c Activity Sheet.

56

Work Session 6 Activity 6c continued

(8.F.4)

Directions: •  Designate one person to show your work

on chart paper and post it.

•  The facilitator will select several groups to report out.

57

Work Session 6 Activity 6c continued

(8.F.4)


Session 6c (item #2 only).

*Note: Work Session 6c has an alternative on the Flash Drive which includes four different graphs.

58

PARCC Model Content Frameworks for

Mathematics

59

PARCC Model Content Frameworks for Mathematics

(page 4)

Purpose: •  Serve as a bridge between the CCSS and

the PARCC Assessments

•  Inform the development of item specifications and the assessment blueprints

60


(page 5) Structure: •  Examples of key advances from the previous grade •  Fluency expectations or examples of culminating

standards •  Examples of major with-in grade dependencies •  Examples of opportunities for connections among

standards, clusters, or domains •  Examples of opportunities for in-depth focus •  Examples of opportunities for connecting

mathematical content and mathematical practices •  Content emphases by cluster

61


(Grade 6 page 27) Example of Key Advances from Grade 5 to Grade 6: Students’ prior understanding of and skill with multiplication, division, and fractions contribute to their study of ratios, proportional relationships and unit rates. (6.RP)

62


Fluency expectations and examples of culminating standards: • Highlight individual standards that set expectations for fluency • Stress the need to provide supports and opportunities for practice

63


(page 8) Fluency: •  Means quickly and accurately •  Means to flow without halting, stumbling, or reversing •  Marks the endpoints of progressions of learning •  Is often an extension of one or more grades earlier in the

standards than the grade when fluency is finally expected •  Does not happen all at once in a single grade •  Requires attention to student understanding along the

way (is not meant to come at the expense of understanding)

64


(Grade 7 page 31)

Fluency Expectations or Examples of Culminating Standards:

In solving word problems leading to one-variable equations of the form px + q = r and p(x+q) = r, students solve the equations fluently. This will require fluency with rational number arithmetic (7.NS.1-3), as well as fluency to some extent with applying properties operations to rewrite linear expressions with rational coefficients (7.EE.1).

65

7.EE.4


(page 11)

Examples of Major Within-Grade Dependencies: • Highlight cases in which a body of content within a given grade depends conceptually or logically upon another body of content within that same grade • Stress the need to organize material coherently within each grade level • Focus only on the large-scale dependencies (coherence is also important for dependencies that exist at finer grain size, but is not described due to space limitations)

66


(Grade 8 page 36) Examples of Major Within-Grade Dependencies:

Much of the work of grade 8 involves lines, linear equations and linear functions (8.EE.5–8; 8.F.3–4; 8.SP.2–3). Irrational numbers, radicals, the Pythagorean theorem and volume (8.NS.1–2; 8.EE.2; 8.G.6–9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a).

67


(page 12) Examples of Opportunities for Connections among Standards, Clusters or Domains: • Highlight opportunities for connecting content in assessments, curriculum, and instruction • Stress the need to avoid approaching the standards as a checklist

68


(Grade 6 page 28) Examples of Opportunities for Connections among Standards, Clusters or Domains: • Students’ work with ratios and proportional relationships (6.RP) can be combined with their work in representing quantitative relationships between dependent and independent variables (6.EE.9). • Plotting rational numbers in the coordinate plane (6.NS.8) is part of analyzing proportional relationships (6.RP.3a, 7.RP. 2) and will become important for studying linear equations (8.EE.8) and graphs of functions. (8.F).

69


(page 12) Examples of Opportunities for In-Depth Focus: • Highlight some individual standards that play an important role in the content at each grade • Stress the need to connect content and practices • Connecting content and practice happens in the context of working problems

70


(Grade 7 page 32) Example of Opportunities for In-Depth Focus:

Work toward meeting this standard builds on the work that led to meeting 6.EE.7 and prepares students for the work that will lead to meeting 8.EE.7.

71

7.EE. 4


Grade 7 (page 33)

Example of Opportunities for Connecting Mathematical Content and Mathematical Practices: When students compare arithmetic and algebraic solutions to the same problem (7.EE.4a), they are identifying correspondences between different approaches (MP.1).

72


(pp 12 – 14) Content Emphases by Cluster: •  Note that not all content in a given grade is

emphasized equally in the standards •  Based on the depth of ideas or the time that they take

to master, some clusters require greater emphasis •  Clusters are designed as Major, Additional, and

Supporting

•  Note that no standard should be neglected because this would leave gaps in student skills and understanding

73


(Grade 6 pp 29-30)

74

Content Emphases by Cluster:

Guidance Regarding the Use of Resources in

Mathematics (pp 8 – 10)

75



(pp 8-10) School districts should consider the following when reviewing existing resources or developing materials:

Materials should: • Align to the CCSS • Foster the Standards for Mathematical Practice • Connect the CCSS and Mathematical Practices • Be mathematically correct • Motivate students • Demand conceptual understanding, procedural skill and fluency, and application • Provide strategies for helping students who have special needs (students with disabilities, English language learners, and gifted students) • Provide strategies for integrating literacy

76


(pp 8-10)

Note: Notice that coverage is not in the aforementioned list. Materials that are excellent but narrow in scope still have value; they can be combined with other like resources and supplemented as needed. Don’t settle for a single mediocre resource that claims to cover all content.

77


(page 65)

78

Additional Note on Modeling (MP.4):


(Appendices)

Appendix A: Lasting achievements in K-8 (p. 66)

Appendix B: Starting points for Transition to the CCSS (p. 68) • Give special attention to how well current materials address the suggested starting points. • Organize implementation work according to progressions.

Appendix C: Rationale for the Grades 3-8 Content Emphases by Cluster (p. 70)

Appendix D: Considerations for College and Career Readiness (p. 74)

79


(page 68) Appendix B: Starting points for Transition to the CCSS:

• Operations and Algebraic Thinking: multiplication and division in grades 3–5, tracing the evolving meaning of multiplication from equal groups and array/area thinking in grade 3 to all multiplication situations in grade 4 (including multiplicative comparisons) and from whole numbers in grade 3 to decimals and fractions in grades 5 and 6.

80


(page 68 - 69) Appendix B: Starting points for Transition to the CCSS: • Number and Operations in Base Ten: multiplication and division in grades 3–6.

• Number and Operations – Fractions: fraction multiplication and division in grades 4–6.

• The Number System: grades 6–7.

• Expressions and Equations: grades 6–8, including how this extends prior work in arithmetic.

81


(page 68 - 69) Appendix B: Starting points for Transition to the CCSS: • Ratio and Proportional Reasoning: its development in grades 6–7, its relationship to functional thinking in grades 6–8, and its connection to lines and linear equations in grade 8.

• Geometry: work with the coordinate plane in grades 5–8, including connections to ratio, proportion, algebra and functions in grades 6–high school.

82


Connections to Assessment: •  The PARCC Assessment System will be

designed to measure conceptual understanding, procedural skill and fluency, and application and problem solving

•  Questions will measure student learning across various mathematical domains and practices

83

Conclusion:

The Impact of CCSS for Mathematics at the

Local Level

84

Next Steps: Considerations and Decisions

•  What strategies must be introduced into classroom instruction?

•  What must happen to encourage conceptual understanding with skills and fluency?

•  How will you (administrator or teacher) have to change?

•  How will you support instruction that must change to meet what is required of students by Common Core assessments?

•  How will your everyday decisions be affected by adjustments that will be considered for the positive effects they will have on student learning?

85

•  Intentional •  Scope and Sequence/Pace

•  Collaborative Planning

•  Student Centered

•  Literacy and Vocabulary Rich

•  Engaging

Instruction

86

CCSS for Mathematics Grades 6-8


“Implementing the CCSS for Mathematics is a journey that each person must take

alone.”

87

Websites and Contact Information

88

Common Core Website www.corestandards.org

MDE website PARCC Website www.mde.k12.ms.us (Hot Topics) www.PARCConline.org

Mississippi Department of Education Office of Curriculum and Instruction

[email protected] (601)359-2586

Marla Davis Office Director II, Mathematics/State Mathematics Specialist

[email protected] (601) 359-3862

*all page references are from this document unless

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